AU SABLE FORKS, N.Y.—Gov. Andrew Cuomo has kept himself far removed this summer from the New York City mayor's race, both literally and figuratively.

Nevertheless, with just several days until Tuesday's primary, Mr. Cuomo stands at the center of perhaps the most heated topic on the campaign trail: frontrunner Bill de Blasio's plan to raise taxes on the wealthy to pay for prekindergarten and after-school programs, a measure that would...